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A Cheeger-type inequality for the drift Laplacian with Wentzell-type boundary condition

Marie Bormann

TL;DR

This paper derives Cheeger-type lower bounds for the first nontrivial eigenvalue of the drift Laplacian on weighted manifolds with Wentzell-type boundary conditions, addressing both sticky reflection without boundary diffusion (δ=0) and with boundary diffusion (δ=1). By formulating the problem via Dirichlet forms and variational principles, it defines bulk-boundary Cheeger constants and proves bounds of the form λ1 ≥ (constants product)/4, with explicit constants capturing interior and boundary interactions. The δ=0 and δ=1 analyses yield complementary inequalities, including λ^{SR}_1 ≥ (overline{h}_B·overline{h}_C)/4 and λ^{SRBD}_1 ≥ ar{h}_D/4, plus refined bounds such as λ^{SRBD}_1 ≥ (min( ilde{h}_C(\Omega), h_C(∂Ω))· ilde{h}_E)/4. The results illuminate how interior versus boundary diffusion and stickiness govern convergence to equilibrium, align with classical Cheeger theory in appropriate limits, and provide tools to estimate spectral gaps in stochastic diffusion models with dynamic boundary behavior.

Abstract

We prove lower bounds for the first non-trivial eigenvalue of the drift Laplacian on manifolds with Wentzell-type boundary condition in terms of some Cheeger-type constants for bulk-boundary interactions. Our results are in the spirit of Cheeger's classical inequality.

A Cheeger-type inequality for the drift Laplacian with Wentzell-type boundary condition

TL;DR

This paper derives Cheeger-type lower bounds for the first nontrivial eigenvalue of the drift Laplacian on weighted manifolds with Wentzell-type boundary conditions, addressing both sticky reflection without boundary diffusion (δ=0) and with boundary diffusion (δ=1). By formulating the problem via Dirichlet forms and variational principles, it defines bulk-boundary Cheeger constants and proves bounds of the form λ1 ≥ (constants product)/4, with explicit constants capturing interior and boundary interactions. The δ=0 and δ=1 analyses yield complementary inequalities, including λ^{SR}_1 ≥ (overline{h}_B·overline{h}_C)/4 and λ^{SRBD}_1 ≥ ar{h}_D/4, plus refined bounds such as λ^{SRBD}_1 ≥ (min( ilde{h}_C(\Omega), h_C(∂Ω))· ilde{h}_E)/4. The results illuminate how interior versus boundary diffusion and stickiness govern convergence to equilibrium, align with classical Cheeger theory in appropriate limits, and provide tools to estimate spectral gaps in stochastic diffusion models with dynamic boundary behavior.

Abstract

We prove lower bounds for the first non-trivial eigenvalue of the drift Laplacian on manifolds with Wentzell-type boundary condition in terms of some Cheeger-type constants for bulk-boundary interactions. Our results are in the spirit of Cheeger's classical inequality.

Paper Structure

This paper contains 3 sections, 7 theorems, 57 equations, 3 figures.

Table of Contents

  1. Connections with classical results
  2. $\delta=0$ - Weighted Brownian motion with sticky reflection
  3. $\delta=1$ - Weighted Brownian motion with sticky reflection and boundary diffusion

Key Result

Lemma 2.1

The eigenvalues of problem eq:wbmwsr form a countably infinite set $\{\lambda^{SR}_{k}\ |\ k\in\mathbb{N}\}\subset \mathbb{R}_+$ without finite accumulation point, and so its elements may be arranged in an increasing sequence and the variational characterisation of these eigenvalues is where $\mathcal{H}_k$ denotes the set of $k$-dimensional subspaces of $H^1(\Omega,\alpha)$.

Figures (3)

  • Figure 1: Exact spectral gap (blue, solid) and lower bound (yellow, dotted) for Euclidean example
  • Figure 2: Exact spectral gap (blue, solid) and lower bound (yellow, dotted) for hyperbolic example
  • Figure 3: Spectral gap for sticky-reflecting boundary diffusion of varying speed $\delta$ (blue, solid) and spectral gap for pure reflection (yellow, dotted) for Euclidean example

Theorems & Definitions (18)

  • Lemma 2.1
  • Lemma 2.2: Comparison of eigenvalues
  • Lemma 2.3: Comparison of Cheeger-type constants
  • proof
  • Theorem 2.1: Cheeger-type inequality
  • proof
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Example 2.1
  • ...and 8 more